Indian sruti scale and discordance

Haresh Bakshi wrote me an e-mail asking about the Indian sruti scale
and applying my favorite discordance measure, harmonic entropy, to it.

In terms of Western music, we may assume (though the keynote will
vary a bit in practice) that a singer will sing in the range G-g',
against drone pitches at the notes C, G, and c.

The notes of Indian music are named as follows:

Sa - c
Ni - Bb-B
Dha - Ab-A
Pa - G
Ma - F-F#
Ga - Eb-E
Ri - Db-D
Sa - C

Except Sa and Pa, which have only a single 'accepted' pitch for each,
the other notes of the heptatonic scale have *four* 'accepted'
pitches (forming the pattern 'Pythagorean'-'Just'-'Just'-
'Pythagorean') each, implying a system of 22 'srutis' per octave.

Here is graph showing every possible sung pitch and its harmonic
entropy against each of the three drone notes, assuming ordinary
harmonic sensitivity:

/tuning/files/dyadic/haresh1.gi
f

This one assumes fine harmonic sensitivity:

/tuning/files/dyadic/haresh2.gi
f

One interesting point is the note A or a, which occupies the third
and fourth (the two closest to Ni) srutis of the note Dha, which is
supposedly most typically played in the higher 'Pythagorean' pitch in
Indian music.

While the lower 'Just' pitch for Dha looks unequivocally more
concordant for the lower octave (A), there is one curve, the green
one, which shows that greater concordance is had in the upper octave
(a) by using the higher 'Pythagorean' pitch -- this being the curve
showing discordance against the Pa (G) drone note.

This may not surprise those who have looked at the ratios. But the
curves show more information, such as the locations of maximally
discordant regions, and perhaps more importantly, the effects of
nonzero tolerance. For example for ordinary harmonic sensitivity, in
the upper octave, the third sruti for Ri (d') looks quite a bit more
concordant against the Pa (G) drone note than most other srutis do --
though it's outranked by its immediate neighbor, the fourth sruti for
Ri. The ratios in this case are 80:27 and 3:1 -- 80:27 showing up as
a lot more concordant than you would expect from the size of the
numbers in the ratio, as Kraig has said of this ratio before.

Enjoy the graphs, and please let me know of any questions (I'll get
to them eventually) . . .

Seasons Greetings,
Paul